4 Answers
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Really, the only equipment you need to gain strength is some floor space and something to hang on. And even the "something to hang on" is, in some people's opinion, optional.

If you can't afford, or don't want to, spend the money on typical gym equipment, then look into the various bodyweight exercises and plans. Convict Conditioning, Never Gymless, You Are Your Own Gym, and Yoga are but some of the things you can do to build strength with nothing but bodyweight.

Berin Loritsch says that you will eventually get to a place where bodyweight alone is insufficient. While he may be technically right, there's a catch - that point comes when you could (if you were doing so) do things like deadlift 2x your bodyweight or more. At which point, you've exhausted such things as muscle-ups, handstand pushups, jumping pistol squats, planches, shrimp squats, l-sits, and back bridges.

I, personally, prefer Yoga, in part because it naturally incorporates things like handstands and planks, so I'm not constantly doing mindless repetitions of things, but rather flowing through things and doing isometrics. About.com has an awesome Yoga section.

There is always a price to becoming stronger. It's a question of how badly do you want to be strong, and what kind of price you are willing to pay. The biggest price you pay has nothing to do with money. The basic principles to getting stronger are:

Progressive overload. If you aren't increasing resistance in some way, you aren't getting stronger.

Recovery. You need to give your body the nutrients and rest it needs to recover from progressive overload and build more muscle.

You can do that in several ways. The quickest and most direct way is to use weights for resistance. This involves barbells, dumbbells, kettlebells, rocks, or anything that weighs a lot--including people. Since you don't want to pay gym fees or buy equipment, you will be using either body weight or unconventional weights for most of your exercises.

Another avenue is to increase resistance. While weights are a direct way to increase resistance, there is also isometric exercises and pushing against something that pushes back--like gravity.

The key here is to find something you can increase in difficulty. For example, if you are doing body weight dips and pull ups, you might look in to developing the strength to do muscle ups (starting with a pull up and transitioning to a dip all in the same movement). Then do it strict, and then slow. You might find some inspiration from Strongman activities as well.

Eventually, you will get to a place where body weight alone is not sufficient. You'll have to break down and start collecting heavy things to lift. This usually involves money, but hopefully by that time you will have an income that will pay your bills and have enough left over to buy the heavy things.

How strong is it possible to become with bodyweight exercises? Amazingly strong. In fact I
would go so far as to say, done correctly, far stronger than someone who had trained for
the same amount of time with free weights. Want some concrete examples? One of my former >students, JJ Gregory (1993 Junior National Champion on the Still Rings) developed such a
high degree of strength from my bodyweight conditioning program that on his first day in >his high school weightlifting class he deadlifted 400lbs., and this at the scale breaking
weight of 135 lbs. and a height of 5'3".

After this I was curious and wanted to measure JJ's one rep max on weighted pull-ups. We
started fairly light with 10 lbs. or so. I continued adding more weight while JJ performed
single rep after single rep. Unfortunately I didn't know about chinning belts and chains at
that time and the cheap leather belt we were using broke at 75 lbs. Once again, I repeat,
at 75 lbs. and JJ had never performed a weighted pull-up in his life. But he had performed
years of my specialized bodyweight conditioning exercises. How much could JJ have chinned
that day? We will never know for sure, but I will tell you that at 75 lbs. JJ was laughing
and joking with me and did not appear to be noticeably bothered by the weight.

And JJ, while the strongest, is not an isolated case. ...

I have been working through this progression for about a month, and while I am still on the first step of the progression, I have already noticed that my core and upper body are quite a bit stronger and am noticing more definition in my arms and chest. I imagine that lower body strength will follow once I move to the next step and my legs are no longer resting on my arms.